Some results on Artinian cofinite top local cohomology modules

Abstract

Let $$(R,\mathfrak {m})$$ be a Noetherian local ring, I be an ideal of R, and M be a finitely generated R-module such that $${\text {H}}_I^t(M)$$ is Artinian and I-cofinite, where $$t={\text {cd}}\,(I,M)$$ . In this paper, we give some equivalent conditions for the property $$\begin{aligned} {\text {Ann}}\,_R\left( 0:_{{\text {H}}_I^t (M)} \mathfrak {p}\right) =\mathfrak {p}~\text {for all prime ideals }~ \mathfrak {p}\supseteq {\text {Ann}}\,_R{\text {H}}_I^t(M).(*) \end{aligned}$$ Also, we show that if $${\text {H}}_I^t(M)$$ satisfies the property $$(*)$$ , then $${\text {H}}_I^t(M)\cong {\text {H}}_{\mathfrak {m}}^t(M/N)$$ for some submodule N of M with $${\text {dim}}\,(M/N)=t$$ .